Contents

For a number n, we define s(n) to be the sum of the
aliquot parts of
n, i.e., the sum of the positive divisors of n, excluding n itself: so,
for example, s(8)=1+2+4=7, and
s(12)=1+2+3+4+6=16. If we start at some number and apply
s repeatedly, we will form a sequence:

s(15)=1+3+5=9,

s(9)=1+3=4,

s(4)=1+2=3,

s(3)=1,

s(1)=0.

If we ever reach 0, we must stop, since all integers divide 0. There are
three obvious possibilities for the behavior of this
aliquot sequence:

A perfect number
is a cycle of length 1 of s, i.e., a number whose positive
divisors (except for itself) sum to itself. For example, 6 is perfect
(1+2+3=6),
and in fact 6 is the smallest perfect number.
The next two perfect numbers are 28 (1+2+4+7+14=28) and 496
(1+2+4+8+16+31+62+124+248=496).

If 2p-1 is prime, then 2p-1(2p-1) is even and perfect,
and conversely, all even perfect numbers have this form. Primes of the form
2p-1 are called Mersenne primes, so we can say that there are just
as many even perfect numbers as Mersenne primes. It is conjectured that the
number of Mersenne primes is infinite; if this is true there will also
be an infinity of even perfect numbers. Also, there are just as many even perfect
numbers known as Mersenne primes known (48 as of 4-II-2015.)

One of the oldest conjectures in mathematics that is still open is that
there are no odd perfect numbers. It has been proved that
any odd perfect number must exceed 10300[BCT],
and must be divisible by a prime power exceeding 1020[COHG1987].

An amicable pair is a cycle of length 2 of s, i.e., a
pair of numbers each of which equals the sum of the
other's aliquot parts; the members of amicable pairs are also called
amicable. The smallest such pair is (220,284).

It is conjectured that there are infinitely many amicable pairs, although,
as for perfect numbers, this is not known.
Here
is a list of all the amicable pairs with lower member below
2.01*1011,
and here is a longer (but less annotated)
list comprising the first 5001 amicable pairs.
For bigger amicable pairs, see [TBBHL], or see the
extensive database below.

The members of aliquot cycles of length greater than 2 are
often called sociable numbers. The smallest two such
cycles have length 5
and 28, and were found early in the last century by Poulet [POU].
Borho [BOR1969] constructed one
of length 4 in 1969. Everything
since has been found via computer search.
Here is a list of all the sociable numbers
I know of.

In the introduction we divided aliquot sequences into three classes.
Catalan [CAT] and
Dickson [DIC] conjectured that all sequences fell into
the first two classes, so that iterating s never approached infinity,
regardless of the starting number.
However, there is now good reason to believe
[GUY1977] that iterating s
does go to infinity for most even starting numbers, although this has not
been proven for any starting number. The smallest starting number whose
sequence has not been completely computed is 276.

We may generalize all these notions in various ways by changing the
criterion we use to say when one number divides another. To start,
we observe that if the prime factorization of n is
p1a1
p2a2 ...
pmam,
then d divides n just when
d can be written in the form
p1c1
p2c2
... pmcm,
where c1 is between 0 and a1,
inclusive,
c2 is between 0 and
a2, inclusive, and so on.

To
generalize the notion of divisibility, we change the relation that the
cis must have to the ais.
If each ci
equals 0 or ai, we call d a unitary
divisor of
n. If each ci is between 0 and
ai, and,
if ai is even, ci does not
equal ai/2, then
d is a bi-unitary divisor of n. If
each
ci divides the corresponding
ai, then d
is called an exponential divisor of n [STR]. If
each
ci+1 divides the corresponding
ai+1, then I call d
a modified exponential divisor of n.
Finally, d is called an infinitary divisor of
n if each ci has a zero bit in its binary
expansion
everywhere that the corresponding ai does
[COHG1990].

After making all these definitions, we can talk about reduced bi-unitary
amicable numbers, augmented infinitary sociable numbers, and so on and
so forth. There are some obvious relations between these notions.
For example, if all members of an aliquot cycle are not divisible by
the square of any prime, or squarefree, it
is also a unitary aliquot cycle. The following table, compiled
February 2015, shows how many
of each of these types of generalized aliquot cycles exist.

Here is a list of
all generalized aliquot cycles, defined as above, with member preceding the
largest member less than 2*1011. The exponential cycles are not
listed, as they can be easily computed from the
modified exponential cycles. Each cycle is preceded by its codes, from
the `Code' column of the table above, and its length. For example, a
line n+b+i+4 would mean that the following cycle is of length 4, and is
an augmented aliquot cycle, augmented bi-unitary aliquot cycle, and
augmented infinitary aliquot cycle.

Jan Otto Munch Pedersen used to have a database of aliquot cycles and generalized aliquot
cycles, Tables of Aliquot Cycles, available on line, at the URL
http://amicable.homepage.dk/tables.htm. It included discoverer information for
each cycle. Unfortunately, it doesn't seem to be available any more,
so I am providing most of the data from this database here.
The data is current as of the last time Pedersen's database was updated, which was on Oct. 1, 2007.

First, if
g is some generalized definition of divisibility, let
Sg(n) be the sum of g-divisors of n.
We have defined generalized divisibility so that if
the prime factorization of n is
p1a1
p2a2 ...
pmam,
then Sg(n)=
Sg(p1a1)
Sg(p2a2) ...
Sg(pmam).
We can now prove various facts we used above.

Now if p is odd, the sum of the modified
exponential divisors of pe will have the same parity
as the number of divisors of e+1, so it will be even unless
e+1 is a square. But if n is augmented or reduced
modified exponential perfect, then Smodified exponential
(n) must be odd, so n must have the prime factorization
2a
p1b1-1 ...
pmbm-1, where a may be
zero, p1,
..., pm are distinct
odd primes, and b1,
..., bm are squares exceeding 1.
Now if a>0,
Smodified exponential(2a)/2a
will be 3/2 if a=1, 5/4 if a=2,
and by the last paragraph, no bigger than 3/2 otherwise. Since each
bi is a square exceeding 1,
each bi is at least 4, so
Smodified exponential(pibi-1)/pibi-1
is no bigger than 7/6 if
pi=3, or (1-pi-2)-5/4
otherwise.
But the product of (1-q-2)-1
over all primes q is the sum of the reciprocals of the squares
of the positive integers, which is pi2/6, so the product
taken over all primes except 2 and 3 is
(pi2/6)/((4/3)(9/8)).
Therefore,
Smodified exponential(n)/n
is no bigger than
(3/2)(7/6)((pi2/6)/((4/3)(9/8)))5/4,
which is less than 1.965, so any reduced modified exponential
perfect number n would have to satisfy 2n+1<
1.965n, which is obviously impossible. Any augmented
modified exponential perfect number n must satisfy
2n-1<1.965n, so n must be one of the positive
integers between 1 and 28. But of all these integers, only 1 and 2
are augmented modified exponential perfect.